Prove that if f XY and g YZ are functions and g f is onetoon

Prove that if f: X->Y and g: Y->Z are functions and g ?f is one-to-one, the f is one-to-one.

gof is one to one implies

gof(x1) = g0f(x2) implies x1 =x2

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Consider f(x1) and f(x2)

If possible let f be not one to one.

Then we have f(x1) =y1, f(x2) =y1 for difference x1 and x2

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gof(x1) = g(y1) = z1

g0f(x2) = g(y1) = z1

This means that there exists two elements x1 and x2 which are not equal such that their images under

gof are equal
A contradiction to gof is one to one.

Hence f is also one to one.